第三章，考虑离散群的混沌作用。
In chapter 4, we consider the ergodic properties of group actions.
对于大量的钮扣，或节点，随着连接线段或边缘线数量的增加，系统突然从离散的断开的小群跳转到一个巨大的连接群。
For a large number of buttons, or nodes, as the number of connecting lines or edges increases, the system rather suddenly jumps from disconnected small clusters to a giant connected cluster.
地下洞室群的交通运输系统是一种典型的离散事件动态系统，难以用解析方法计算的数学模型来描述。
The traffic transportation system of underground structure group is a kind of typical discrete event dynamic system, which is difficult to describe with analytic mathematic models.
第二种方法就是运用离散对数的概念。要掌握这些概念首先需要了解乘法群的概念。
The second approach is to use the concept of discrete logarithm. Understanding this concept requires understanding some properties of multiplicative groups.
文中提出了一个新的群签名方案，该方案的安全性基于离散对数困难问题，其运行效率比已有的同类方案高。
A new group signature scheme based on the discrete logarithm problem is presented. The scheme is more efficient than the previously known schemes with the same security properties.
方法在NO扩散的点源模型基础上，构建了一个离散化神经元群的平板模型。
Method Based on point model regarding a single neuron as a point source, a discrete plat model of NO diffusion was established.
通过引入随机交换序、PMX算子使微粒群优化算法能够求解车辆路径问题这类离散组合优化问题。
PSO can solve a discrete combination optimization such as VRP by using random exchange sequence and PMX operator.
针对蚁群算法在求解连续优化问题上相对较弱的特点，提出了基于网格划分的蚁群算法，将传统的用于求解离散空间优化问题的蚁群算法进行了扩展。
Puts forward a new ACA based on grid division to overcome the weakness of ACA in solving continuous function optimization problem, and expands the discrete space to the continuous space.
提出了一种基于微粒群优化(PSO)算法的连续属性离散化方法，很好的解决了建模过程中连续属性的离散化问题。
An algorithm for discretization based on Particle swarm optimization (PSO) is presented, which can settle the problem of continuous attributes discretization in systema modeling perfectly.
通过引入随机交换序、PMX算子使微粒群优化算法能够求解车辆路径问题这类离散组合优化问题。
A PSO can solve a discrete combination optimization such as VRP by using random exchange sequence and PMX operator.
将土与群桩体系视为一个整体进行有限元离散，采用等效线性化方法考虑土体的动力非线性性能。
The interaction system as an integrality is modeled by some finite elements and the dynamic nonlinearity of soils is considered through the equivalent linearity method.
借鉴蚁群算法的信息素机制，提出了一种基于信息素机制的离散粒子群算法。
A pheromone-based discrete particle swarm optimization algorithm was proposed borrowing the idea of pheromone refresh mechanism of ant colony algorithm.
在离散微粒群算法的基础上，提出了一种基于二进制的随机多目标P SO算法，并对感知模型进行覆盖优化。
Then a binary awareness model based on stochastic sensor placement and a stochastic multi-objective PSO arithmetic based on binary system which is applied on awareness model have been present.
蚁群算法是基于群体合作的一类仿生算法，适合于解困难的离散组合优化问题。
Ant colony algorithm is a novel simulated evolutionary algorithm based on group cooperation and can be applied to solve hard discrete combinatorial optimization problem.
本文研究了一类三种群离散型捕食者食饵系统。
In this paper, a class of predator-prey systems of three species with discrete time is studied.
利用群逆可以求得离散型对称奇异系统的显解。
Using the group inverse, an explicit solution to a symmetric singular system is described.
讨论一类可数离散半群上概率测度卷积幂的弱收敛性，主要结果是利用局部群化的观点给出了概率测度卷积幂弱收敛的一个充分条件。
The main result is that we get a sufficient condition for the weak convergence of convolution powers of probability measures, by using the method of local grouplization.
将离散单元法(DEM)拓展到三维流固两相流的数值模拟中，提出了颗粒群轨道柔性模型。
Let the discrete element method (DEM) be developed into the numerical simulation of 3d fluid-solid two-phased flows, the trajectory flexible model of particle group has been put forward.
由于求解该离散对数的难度与求解乘法群上的离散对数的难度相同，所以在椭圆曲线上可构造出相对安全的密码系统。
Because computing the discrete logarithm more difficulty than the discrete logarithm over multiplication group, so it is possible to construct quite safe cryptosystem.
为解决工程应用中三维流固两相流动力学建模的难题，提出了基于离散单元法(dem)的颗粒群轨道柔性模型。
For dynamics modeling of 3d fluid-solids two-phase flows in engineering problem it was brought forward of particles trajectory flexible model based on the discrete element method (DEM).
为了解决多重模态最优化问题，我们运用了一维离散优化方法、遗传算法和蚁群算法。
To solve the multimodal optimization problem the 1d discrete optimization methods, the genetic and Ant Colony algorithms are applied.
蚁群算法是一种模拟群体智能的算法，在解决基于离散空间的问题时表现出良好的性能。
Ant Colony algorithm is a kind of algorithm that simulates swarm intelligence. It has a good performance in solving the problems based on Discrete Space.
讨论了离散域蚁群算法和连续域蚁群算法的异同。
Similarities and differences between the ant colony algorithm of discrete filter and ant colony algorithm of continuous domain are discussed.
讨论了离散域蚁群算法和连续域蚁群算法的异同。
Similarities and differences between the ant colony algorithm of discrete filter and ant colony algorithm of continuous domain are discussed.
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